Golden Integers

The positive root of the golden polynomial is the famous golden ratio and the negative root is . There are two integer solutions and to the congruence , where is a prime and is a positive integer (in this Demonstration, and ). Can either or be called a "golden integer" that is analogous to ?

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Kurt Hensel (1861–1941) discovered that for certain prime numbers , there are two roots to the "golden polynomial" modulo . Further, he claimed that for each of those primes and for every natural number , there would also exist two roots congruent modulo .

When you choose the "base " option and vary from 1 to 50, it is clear that the 50 different congruences are related. They are a single infinite sequence written as (base ) that is truncated on the left to leave significant digits . Likewise, the 50 checks are a single infinite integer (base ) that is truncated to digits.

With the "base " option set and , notice the strong analogy between modular arithmetic and real arithmetic as you vary . The infinite sequences (base ) behave like "real" infinitely repeating decimals. And like repeating decimals, the most significant digits are nearer the decimal point.

Writing the 50 congruences as a single truncated sequence (base ) inspired the introduction of -adic numbers by Kurt Hensel in 1897. Powerful applications of -adic numbers have been found in number theory, including the famous proof of Fermat's last theorem by Andrew Wiles.

In conclusion, we have found two infinite sequences of integer roots (base ) to the "golden polynomial" for a given radix . But alas, it is impossible to determine which integer root, or is "positive" and can receive the name "the golden -adic integer" for radix .